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CHAPTER 1 COMPLEX NUMBERS
THE MODULUS AND ARGUMENT THE POLAR FORM THE EXPONENTIAL FORM DE MOIVRE’S THEOREM FINDING ROOTS
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Argument of Complex Numbers
Definition 1.7 The argument of the complex number z = a + bj is defined as 1st QUADRANT 2nd QUADRANT 4th QUADRANT 3rd QUADRANT
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Argument of Complex Numbers
Example 1.8 : Find the arguments of z:
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THE POLAR FORM OF COMPLEX NUMBER
(a,b) r Re(z) Im(z) Based on figure above:
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The polar form is defined by:
Example 1.9: Represent the following complex number in polar form:
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Example 1.10 : Express the following in standard form of complex number:
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Theorem 1: If z1 and z2 are 2 complex numbers in polar form where then,
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Example 1.11 : If z1 = 2(cos40+jsin40) and z2 = 3(cos95+jsin95) . Find : If z1 = 6(cos60+jsin60) and z2 = 2(cos270+jsin270) . Find :
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THE EXPONENTIAL FORM DEFINITION 1.8
The exponential form of a complex number can be defined as Where θ is measured in radians and
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THE EXPONENTIAL FORM Example 1.15
Express the complex number in exponential form:
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THE EXPONENTIAL FORM Theorem 2 If and , then:
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THE EXPONENTIAL FORM Example 1.16 If and , find:
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DE MOIVRE’S THEOREM Theorem 3 If is a complex number in polar form to any power of n, then De Moivre’s Theorem: Therefore :
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DE MOIVRE’S THEOREM Example 1.17 If , calculate :
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FINDING ROOTS Theorem 4 If then, the n root of z is: (θ in degrees) OR (θ in radians) Where k = 0,1,2,..n-1
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FINDING ROOTS Example 1.18 If then, r =1 and : Let n =3, therefore k =0,1,2 When k =0: When k = 1:
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FINDING ROOTS When k = 2: Sketch on the complex plane: y
1 y x nth roots of unity: Roots lie on the circle with radius 1
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FINDING ROOTS Example 1.18 If then, and : Let n =4, therefore k =0,1,2,3 When k =0: When k = 1:
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FINDING ROOTS Let n =4, therefore k =0,1,2,3 When k =2: When k = 3:
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FINDING ROOTS Sketch on complex plane: y x
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